Revisiting Gauss's analogue of the prime number theorem for polynomials over a finite field
Let \(p\) be a prime number. The nonnegative integers are in bijection with the one-variable polynomials over the finite field \({\mathbb F}_p\) by : \[ a_np^n+a_{n-1}p^{n-1}+\dots+a_1p+a_0\leftrightarrow a_nT^n+a_{n-1}T^{n-1}+\dots+a_1T+a_0, \] where the integer represented on the left side is written in base \(p\). Denote by \(\|A\|=a\) the integer \(a\) which corresponds to the polynomial \(A\). For \(X>0\) let \[ \pi_p(X):=\#\{P\in{\mathbb F}_p[T]:\|P\|<X,P\,\text{irreducible}\}, \] \[ \text{ls}_p(X):=\sum_{\substack{\|f\|<X\\ \deg f>0}}\frac{1}{\deg f}. \] The author proves: Theorem 1. If \(X\geq p\), \(p^n\leq X<p^{n+1}\) then \[ \pi_p(X)=\text{ls}_p(X)+O(np^{\frac{n}{2}+1}). \] Theorem 2. If \(X\geq p^2\), \(p^n\leq X<p^{n+1}\), \(r\geq 2\) then \[ \pi_p(X)=\frac{X}{n}+\sum_{k=2}^r(1-\frac{1}{p})A_{p,k}\frac{p^n}{n^k}+O\left(np^{\frac{n}{2}+1}+A_{p,r+2}\frac{p^n}{n^{r+1}}+\frac{p}{n}\sum_{k=1}^rA_{p,k}\right), \] with an absolute \(O\)-constant and \[ A_{p,k}:=\sum_{m=1}^\infty\frac{m^{k-1}}{p^{m-1}}. \] For \(r=2\) one obtains \[ \pi_p(X)\sim \frac{X}{\log_pX},\quad \log_p(X)\to\infty, \] which should be viewed as a version of the prime numnber theorem for polynomials. Similar results can be proved over a general finite base field.
scientific article; zbMATH DE number 5731598